CHAPTER XXV DISTORTION OF WAVE-SHAPE AND ITS CAUSES 232. In the preceding chapters we have considered the alter- nating currents and alternating e.m.fs. as sine waves or as replaced by their equivalent sine waves. While this is sufficiently exact in most cases, under certain circumstances the deviation of the wave from sine shape becomes of importance, and with certain distortions it may not be pos- sible to replace the distorted wave by an equivalent sine wave, since the angle of phase displacement of the equivalent sine wave becomes indefinite. Thus it becomes desirable to investi- gate the distortion of the wave, its causes and its effects. Since, as stated before, any alternating wave can be repre- sented by a series of sine functions of odd orders, the inves- tigation of distortion of wave-shape resolves itself in the in- vestigation of the higher harmonics of the alternating wave. In general we have to distinguish between higher harmonics of e.m.f. and higher harmonics of current. Both depend upon each other in so far as with a sine wave of impressed e.m.f. a distorting effect will cause distortion of the current wave, while with a sine wave of current passing through the circuit, a dis- torting effect will cause higher harmonics of e.m.f. 233. In a conductor revolving with uniform velocity through a uniform and constant magnetic field, a sine wave of e.m.f. is generated. In a circuit with constant resistance and constant reactance, this sine wave of e.m.f. produces a sine wave of current. Thus distortion of the wave-shape or higher har- monics may be due to lack of uniformity of the velocity of the revolving conductor; lack of uniformity or pulsation of the magnetic field; pulsation of the resistance or pulsation of the reactance. 341 342 ALTERNATING-CURRENT PHENOMENA The first two cases, lack of uniformity of the rotation or of the magnetic field, cause higher harmonics of e.m.f. at open circuit. The last, pulsation of resistance and reactance, causes higher har- monics only when there is current in the circuit, that is, underload. Lack of uniformity of the rotation is hardly ever of practical interest as a cause of distortion, since in alternators, due to mechanical momentum, the speed is always very nearly uniform during the period. A periodic pulsation of speed may occur in low speed singlephase machines. Thus as causes of higher harmonics remain: 1st. Lack of uniformity and pulsation of the magnetic field, causing a distortion of the generated e.m.f. at open circuit as well as under load. 2d. Pulsation of the reactance, causing higher harmonics under load. 3d. Pulsation of the resistance, causing higher harmonics under load also. Taking up the different causes of higher harmonics, we have : Lack of Uniformity and Pulsation of the Magnetic Field. 234. Since most of the alternating-current generators con- tain definite and sharply defined field-poles covering in different types different proportions of the pitch, in general the mag- netic flux interlinked with the armature coil will not vary as a sine wave, of the form $ cos /3, but as a complex harmonic function, depending on the shape and the pitch of the field-poles and the arrangement of the armature conductors. In this case the magnetic flux issuing from the field-pole of the alternator can be represented by the general equation, •S* = Ao + Ai cos/3 + A2COS2/3 + AsCosS/S -I- . . . + Bi sin iS + B2 sin 2 /3 + B3 sin 3 /3 + . . . If the reluctance of the armature is uniform in all directions, so that the distribution of the magnetic flux at the field-pole face does not change by the rotation of the armature, the rate of cutting magnetic flux by an armature conductor is <&, and the e.m.f. generated in the conductor thus equal thereto in wave-shape. As a rule A^, Ai, A4 . . . B^, B^ equal zero; that is, successive field-poles are equal in strength and distribu- DISTORTION OF WA VE-SHAPE AND ITS CA USES 343 tion of magnetism, but of opposite polarity. In some types of machines, however, especially inductor alternators, this is not the case. The e.m.f. generated in a full-pitch armature turn — that is, armature conductor and return conductor distant from former by the pitch of the armature pole (corresponding to the distance from field-pole center to pole center) — is 5e = 4>o — $180 = 2 {Aicos/? + ^3Cos3)8 + A5Cos5|8 + . . . + 5i sin i3 + ^3 sin 3 /3 + ^5 sin 5 jS -1- . . . } 110 N ) Lo id ,^ ^ -^ s. 130 x.= = t4 .5 yi = 2,6 /; / s N, 120 .y / N ^^ 110 ^^ 100 / N \ 90 / f \ 80 / >^v 70 / / GO / \ 50 / / \ 10 / ^ 30 k \ 20 /' ^ 10 // 0 '^ -^ , I-IO 0 10 20 30 to 50 60 70 80 90 100 110 120 TSo 110 150 160 170 1 180 Fig. 172. Even with an unsymmetrical distribution of the magnetic flux in the air-gap, the e.m.f. wave generated in a full-pitch armature coil is symmetrical, the positive and negative half- waves equal, and correspond to the mean flux distribution of adjacent poles. With fractional pitch- windings — that is, wind- ings whose turns cover less than the armature pole-pitch — the generated e.m.f. can be unsymmetrical with unsymmetrical magnetic field, but as a rule is symmetrical also. In unitooth alternators the total generated e.m.f. has the same shape as that generated in a single turn. With the conductors more or less distributed over the surface of the armature, the total generated e.m.f. is the resultant of 344 ALTERNATING-CURRENT PHENOMENA several e.m.fs. of different phases, and is thus more uniformly varying; that is, more sinusoidal, approaching sine shape to within 3 per cent, or less, as for instance the curves Fig. 172 and Fig. 173 show, which represent the no-load and full-load wave of e.m.f, of a three-phase multitooth alternator. The prin- cipal term of these harmonics is the third harmonic, which con- sequently appears more or less in all alternator waves. As a rule these harmonics can be considered together with the har- monics due to the varying reluctance of the magnetic circuit. In iron-clad alternators with few slots and teeth per pole, the passage of slots across the field-poles causes a pulsation of the 130 W itnl oad '^— 120 X, = 12 7,0 y,- 3.2 / -~- ^ V no / \ 100 / \ 90 , ^ 80 /■ V 70 / \ 60 / \ t 50 / \ 40 / f H V 30 / \ 30 / \ 10 /i '^\ 0 // ^^ ^ ^ -^ ^ ■ 10 0 10 20 30 10 ^ M 60 70 80 90_ 100 no 120 130 110 150 160 170 180 Fig. 173. magnetic reluctance, or its reciprocal, the magnetic reactance of the circuit. In consequence thereof the magnetism per field- pole, or at least that part of the magnetism passing through the armature, will pulsate with a frequency 2 7, if 7 = number of slots per pole. Thus, in a machine with one slot per pole the instantaneous magnetic flux interlinked with the armature conductors can be expressed by the equation where and <(> = <^ cos /? { 1 + e cos [2 /3 - = average magnetic flux, e = amplitude of pulsation, 6 = phase of pulsation. e] DISTORTION OF WA VE-SHAPE AND ITS CA USES 345 In a machine with 7 slots per pole, the instantaneous flux inter- linked with the armature conductors will be 0 = cos /3 { 1 + e cos [2 7,3 - ^] } , if the assumption is made that the pulsation of the magnetic flux follows a simple -sine law, as first approximation. In general the instantaneous magnetic flux interlinked with" the armature conductors will be 0 = $ cos /3 { 1 + ei cos (2 /3 - ^i) + 62 cos (4 /3 - do) + \ J ' where the term e^ is predominating, if 7 = number of armature slots per pole. This general equation includes also the effect of lack of uniformity of the magnetic flux. In case of a pulsation of the magnetic flux with the fre- quency, 2 7, due to an existence of 7 slots per pole in the arma- ture, the instantaneous value of magnetism interlinked with the armature coil is 0 = $ cos /3 ( 1 -f e cos [2 7/3 - e] ) . Hence the e.m.f. generated thereby, d(j) = - V2Trf^^ {coS|S(l + e cos [2 7^ - 0])}' And, expanded, e = V2 x/n j sin /3 + e " ^y— sin [(27- 1)13 - 6] + 6^^^^sin[(27+l)^-^] }• Hence, the pulsation of the magnetic flux with the frequency, 2 7, as due to the existence of 7 slots per pole, introduces two harmonics, of the orders (2 7 — 1) and (2 7 + 1). 235. If 7 = 1 it is e = V2 7r/n$ { sin /3 + | sin (/3 - 0) + ~ sin (3/3-0) j ; that is, in a unitooth single-phaser a pronounced triple har- monic may be expected, but no pronounced higher hai'monics. Fig. 174 shows the wave of e.m.f. of the main coil of a mono- cyclic alternator at no load, represented by, 346 ALTERNATING-CURRENT PHENOMENA e = ^ { sin /3 - 0.242 sin (3 /3 - 6.3) - 0.046 sin (5 ^3 - 2.6) + 0.068 sin (7 ,8 - 3.3) - 0.027 sin (9 iS - 10.0) - 0.018 sin (11 ^ - 6.6) + 0.029 sin (13 /? - 8.2)}; hence giving a pronounced triple harmonic only, as expected. If 7 = 2, it is, e = \/2 7r/n j sin /3 + -^ sin (3 /3 - 0) + i^%in (5 /3 - ^) [ , the no-load wave of a unitooth quarter-phase machine, having pronounced triple and quintuple harmonics. 120 110 100 90 80 70 60 so 40 so 20 10 0 .10 . N f :r^ "== '^, ,^ 7' > V r .-ll ^ ^ \ _4 ^2^. :<^ er\l > 7 ^ \, J \ ■--. ■" l^srrTair.d jrof Ui hit ^ P" \ Kj ■y" 0 10' 20" 30° 40° S0° 60° 70° 80° 90° 100° "110° 120° 13tfl40' 150° 160° 170° W Fig. 174. — No-load of e.m.f. of unitooth monocyclic alternator. If 7 = 3, it is, e = \/2irfn^\ sin ^3 + -^ sin (5 ^3 — ^) + -^ sin (7 /3 — / That is, in a unitooth three-phaser, a pronounced quintuple and septuple harmonic may be expected, but no pronounced triple harmonic. Fig. 175 shows the wave of e.m.f. of a unitooth three-phaser at no-load, represented by e = E {sin 0 - 0.12 sin (3 /S - 2.3) - 0.23 sin (5 /3 - 1.5) -I- 0.134 sin (7 i3 - 6.2) - 0.002 sin (9 /S + 27.7) - 0.046 sin (11 |8 - 5.5) + 0.031 sin (13 /3 - 61.5)). Thus giving a pronounced quintuple and septuple and a DISTORTION OF WA VE-SHAPE AND ITS CA USES 347 lesser triple harmonic, probably due to the deviation of the field from uniformity, as explained above, and deviation of the pulsation of reluctance from sine-shape. In some especially favorable cases, harmonics as high as the 35th and 37th have been observed, caused by pulsation of the reluctance, and even still higher harmonics. In general, if the pulsation of the magnetic reactance is denoted by the general expression 1 + 2^€^r cos (2 7/3 - 9^, 140 130 120 110 100 90 80 70 60 50 40 30 20 10 -J^^ — \ \ \ A \ B \ J \ V / 1 d .o V ^7 4 \ / \ 1, / \ \ / \ / \ ^ c \ ^ . ----. =^ — 0 -^ '' f zr Jp^'^ ° < V ~H ghe \r X 0 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130°140'150160'170180° Fig. 175. — No-load wave of e.m.f. of unitooth three-phase alternator. the instantaneous magnetic flux is <^ = $ cos /3 I 1 + Sye^ cos (2 7/3 - 9^) I 1 - $ j cos ^ + ^' cos (/3 - ^0 + 27 b: cos [(2 7 + 1) [2 1 L2 iS - ^J + %■ cos [(2 7 + l)/3 'Ih hence, the e.m.f., e = \/2 7r/n4> sin /3 + ~ sin (/3 - d^ + St ^ ^^ ^ [2 1 2 [e, sin [(2 7 + l)/3 - ^,] + e,,, sin [(2 7 + l)/3 - 0,,J] 348 ALTERNATING-CURRENT PHENOMENA With the general adoption of distributed fractional pitch arma- ture windings, such pronounced wave shape distortions as shown by the unitooth alternators shown as illustrations, have become infrequent. Pulsation of Reactance 236. The main causes of a pulsation of reactance are mag- netic saturation and hysteresis, and synchronous motion. Since in an iron-clad magnetic circuit the magnetism is not propor- tional to the m.m.f., the wave of magnetism and thus the wave of e.m.f. will differ from the wave of current. As far as this distortion is due to the variation of permeability, the distortion is symmetrical and the wave of generated e.m.f. represents no power. The distortion caused by hysteresis, or the lag of the magnetism behind the m.m.f., causes an unsymmetrical distor- tion of the wave which makes the wave of generated e.m.f. differ by more than 90° from the current wave and thereby represents power — the power consumed by hysteresis. In practice both effects are always superimposed; that is, in a ferric inductive reactance, a distortion of wave-shape takes place due to the lack of proportionality between magnetism and m.m.f. as expressed by the variation in the hysteretic cycle. This pulsation of reactance gives rise to a distortion con- sisting mainly of a triple harmonic. Such current waves dis- torted by hysteresis, with a sine wave of impressed e.m.f., are shown in Figs. 80 and 81, Chapter XII, on Hysteresis. In- versely, if the current is a sine wave, the magnetism and the e.m.f. will differ from sine-shape. For further discussion of this distortion of wave-shape by hysteresis. Chapter XII may be consulted. 237. Distortion of wave-shape takes place also by the pul- sation of reactance due to synchronous rotation, as discussed in the chapter on Reaction Machines, in "Theory and Calculation of Electrical Apparatus." With a sine wave of e.m.f., distorted current waves result. Inversely, if a sine wave of current, i = I cos jS, exists through a circuit of synchronously varying reactance, as for instance, the armature of a unitooth alternator or syn- chronous motor — or, more general, an alternator whose arma- DISTORTION OF WA VE-SHAPE AND ITS CA USES 349 ture reluctance is different in different positions with regard to the field-poles — and the reactance is expressed by X ^ X {1 -\- ecos{2 ^ - d)}; or, more general, X = x{ 1 + X^ey cos (2 7/3 - dj 1 the wave of magnetism is X X I "^ 1 = ^-T- cos /3 = — — - cos i3 + XyC^ cos /3 cos (2 7/3- 9^) \ 2 7r/7i 2 x/n [ 1 -^ I cos ^ + '-' COS (^ - 90 + 2, 2 7r/n [2 1 I COS [(2 7 + 1) €2 J J /3 - g + —^-^' cos [(2 7 + 1) i3 - ^^ + i] hence the wave of generated e.m.f., e = — n-j- — — 2 TTjn -j- dt •' dl3 = a; sin ^ + ^^ sin (^ - 9,) + 2t ^ ^^+ ^ [e^ sin [(2 7 + 1) ^ - g + 62 ^,, sin [(2 y + 1)13 - d^ + 1]] ; that is, the pulsation of reactance of frequency, 2 7, introduces two higher harmonics of the order (2 7 — 1) and (2 7 + 1). If X = x{l -h ecos (2/3 - d)}, it is = 2^{ cos 1(3 + I cos (^ - ^) + |cos (3 /3 - 0) } ; e = a: j sin i3 +^ sin (0 - 9) + -^ sin (3 /3 - 0) • Since the pulsation of reactance due to magnetic saturation and hysteresis is essentially of the frequency, 2 / — that is, describes a complete cycle for each half-wave of current — this shows why the distortion of wave-shape by hysteresis consists essentially of a triple harmonic. The phase displacement between e and i, and thus the power consumed or produced in the electric circuit, depends upon the angle, 9, as discussed before. 350 ALTERNATING-CURRENT PHENOMENA 238. In case of a distortion of the wave-shape by reactance, the distorted waves can be replaced by their equivalent sine waves, and the investigation with sufficient exactness for most cases be carried out under the assumption of sine waves, as done in the preceding chapters. Similar phenomena take place in circuits containing polari- zation cells, leaky condensers, or other apparatus representing a synchronously varying negative reactance. Possibly dielectric hysteresis in condensers causes a distortion similar to that due to magnetic hysteresis. Inversely, at very high voltages, where corona appears on the conductors, with a sine wave of impressed voltage, a distor- tion of the capacity current wave occurs, which is largely effect- ive, but partly reactive due to the increase of capacity under corona. Pulsation of Resistance 239. To a certain extent the investigation of the effect of synchronous pulsation of the resistance coincides with that of reactance; since a pulsation of reactance, when unsymmetrical with regard to the current wave, introduces a power component which can be represented by an "effective resistance." Inversely, an unsymmetrical pulsation of the ohmic resistance introduces a wattless component, to be denoted by "effective reactance." A typical case of a synchronously pulsating resistance is represented in the alternating arc. The apparent resistance of an arc depends upon the current through the arc; that is, the apparent resistance of the arc = potential difference between electrodes . , . . , „ ■ IS high for small currents, current low for large currents. Thus in an alternating arc the apparent resistance will vary during every half-wave of current between a maximum value at zero current and a minimum value at maxi- mum current, thereby describing a complete cycle per half-wave of current. Let the effective value of current through the arc be repre- sented by /. Then the instantaneous value of current, assuming the current wave as sine wave, is represented by i = I -v/2 sin |3; DISTORTION OF WA VE-SHAPE AND ITS CA USES 351 and the apparent resistance of the arc, in first approximation, by R = r(l -\- ( cos 2/3); thus the potential difference at the arc is e = iR = I\/2r sin /3 (1 + « cos 2 0) = rIs/2 j ( 1 - 0 sin /8 + I sin 3 ^ Hence the effective value of potential difference, '1 - . + I'. and the apparent resistance of the arc, E €2 U = j = ry^\-. + ^- The instantaneous power consumed in the arc is ie = 2 rP j (l - ^) sin^ /3 + | sin /3 sin 3 iS } • Hence the effective power, p = .p(i-|). The apparent power, or volt-amperes consumed by the arc, IE = rP-^jl - e + ^'- Thus the power-factor of the arc, _ P^ _ 2 p-iE- I r ' Vl-^ + 2 that is, less than unity. 240. We find here a case of a circuit in which the power-factor — that is, the ratio of watts to volt-amperes — differs from unity without any displacement of phase; that is, while current and e.m.f. are in phase with each other, but are distorted, the alter- nating; wave cannot be replaced by an equivalent sine wave. 352 ALTERNATING-CURRENT PHENOMENA since the assumption of equivalent sine wave would introduce a phase displacement, cos d = p of an angle, 9, whose sign is indefinite. As an example are shown, in Fig. 176, for the constants, / = 12, r = 3, e = 0.9, the resistance, i? = 3 (1 + 0.9 cos 2^); the current, i = 17 sin /3; f \ -A r / \ ^ / ^ -~\ f \ e "^ /^ "Nj / \ \ / \^ / / \ r > ^R \, 1 / \ >v \, 1 / > \ \ \ / / y / K ^ > r / \ \ / / / A \ / \ ^_ / / — / A / \ /' !/ f— \ / A \ r ' — \ 'AF lA 3.LE R :sij 3t;i NC i / R = 3 1 +. 9c ,r. ^) / \ 1 1 i = 17 sin /3 \ / \ y J e = 28( iin^+. i2 s n 3 ^) V. / ^ J 1 Fig. 176. — Periodically varying resistance. the potential difference, e = 28 (sin /3 + 0.82 sin 3 /3). In this case the effective e.m.f. is E = 25.5; the apparent resistance, the power, the apparent power, the power-factor. ro = 2.13; P = 244; EI = 307; p = 0.796. DISTORTION OF WA VE-SHAPE AND ITS CA USES 353 As seen, with a sine wave of current the e.m.f. wave in an alternating arc will become double-peaked, and rise very abruptly near the zero values of current. Inversely, with a sine wave of e.m.f. the current wave in an alternating arc will become peaked, and very flat near the zero values of e.m.f. 241. In reality the distortion is of more complex nature, since the pulsation *of resistance in the arc does not follow a simple sine law of double frequency, but varies much more abruptly near the zero value of current, making thereby the variation of e.m.f. near the zero value of current much more abruptly, or, inversely, the variation of current more flat. 0 8 \- ^" "■" — J? 2 i 1 F t .■^ > I 6 B i \ H'^ ^ 4^ ^- 1 1 ,I.J\1 \ ^ / — * ^ S "* V / ^ s E 1 » I S / \ / \ , / \l \> ) •^ h 1 1 1 1 i»i } in 11 12 13 I'l 1B°| 16 17 1 8 19 20 21 2 1 [H 2 23 24' '25 V f^ONE PAIR CARBONS iEGULATEDBYHAND 1. A. C. dynamo e. m. f. 1. •' " " currenti. i / \ / ' - \ / V -! , 1— « rA 5 J ^f~ / Y] / 1 1 ■\ \ / \ 6 / X 1 y' / \ f)^ \ ■ 'i 8 ff _J _ _ _ Lj _ Fig. 177. — Electric arc. A typical wave of potential difference, with an approximate sine wave of current through the arc, is given in Fig. 177.^ 242. The value of e, the amplitude of the resistance pulsation, largely depends upon the nature of the electrodes and the steadiness of the arc, and with soft carbons and a steady arc is small, and the power-factor, p, of the arc near unity. With hard carbons and an unsteady arc, e rises greatly, higher harmonics appear in the pulsation of resistance, and the power-factor, p, falls, being in extreme cases even as low as 0.6. Especially is this the case with metal arcs. This double-peaked appearance of the voltage wave, as ,shown by Figs. 176 and 177, is characteristic of the arc to such an extent ^ From American Institute of Electrical Engineers, Transactions, 1890, p. 376. Tobey and Walbridge, on the Stanley Alternate Arc Dynamo. 23 354 ALTERNATING-CURRENT PHENOMENA that when in the investigation of an electric circuit by oscillo- graph such a wave-shape is found, the existence of an arc or arcing ground somewhere in the circuit may usually be sus- pected. This is of importance as in high-voltage systems arcs are liable to cause dangerous voltages. The pulsation of the resistance in an arc, as shown in Fig. 177 for hard carbons, is usually very far from sinusoidal, as assumed in Fig. 176. It is due to the feature of the arc that the voltage consumed in the arc flame decreases with increase of current — approximately inversely proportional to the square root of the current — and so is lowest at maximum current. Approximately, the volt-ampere characteristic of the arc can be represented by, c e = eo -\ -^, (1) V t where eo is a constant of the electrode material (mainly), c a con- stant depending also upon the electrode material and on the arc length, and approximately proportional thereto. This equation would give e = oo, for i = 0. This obviously is not feasible. However, besides the arc conduction as given by above equation — which depends upon mechanical motion of the vapor stream — a slight conduction also takes place through the residual vapor between the electrodes, as a path of high resistance, r, and near zero current, where the voltage is not sufficient to maintain an arc, this latter conduction carries the current. The characteristic of the alternating-current arc therefore consists of the combination of two curves : the arc characteristic, (1), and the resistance characteristic, e = n. (2) The phenomenon then follows' that curve which gives the lowest voltage ; that is, for high values of current, is represented by equation (1), for low values of current, by equation (2). 243. As an example are shown in Fig. 178 the calculated curves of an alternating arc between hard carbons (or carbides), for the constants, eo = 30 volts, c = 40, r = 70 ohms. DISTORTION OF WA VE-SHAPE AND ITS CA USES 355 The curve I represents the arc conduction, following equation (1), e = 30 + ^,, V i and the curve II represents the conduction through the (sta- tionary) residual vapor, by equation (2), near the zero points, A and D, of the current, e = 70 i. As seen, from A to B the voltage varies approximately pro- portionally with the current. At B the arc starts, and the vol- 1 — 1 , 1 \ J f 1 1 \ _ _i / S, By f \ n - } s > ' \ ^ / ' 1 S s^ I ^ w 11 /ii / N I " Y / \J / \ 1 a/ \Id" / K / 1 \ \ / 1 / I \ / /{ , k / s^ j 1 y V — " / -l > / > y F eK \ / 1 / \ / > 1 \ 1 "■■/ 1 ' Fig. 178. tage drops with the further increase of current, and then rises again with the decreasing current, until at C, the intersection point between curves I and II, the arc extinguishes and the voltage follows curve II, until at E the arc starts again. The two sharp peaks of the curve thus represent respectively the starting and the extinction of the arc. Since the high values of voltage near zero current lower and the low values of voltage near maximum current raise the value of 356 ALTERNATING-CURRENT PHENOMENA the current, the current wave does not remain a sine wave, if the arc voltage is an appreciable part of the total voltage, but the current wave becomes peaked, with flat zero, as expressed approximately by a third harmonic in phase with the funda- mental. The current wave in Fig. 178 so has been assumed as i = 13 cos 0 + 2 cos 3 0. From Fig. 178 follows: effective value of current, 9.30 amp., effective value of voltage, 47.2 volts; hence, volt-amperes consumed by the arc, 439 volt-amp. ; and, by averaging the products of the instantaneous values of volts and amperes, power consumed in the arc, 388 watts; hence, power-factor, 77 per cent. If the resistance, ?*, of the residual arc-vapor is lower, as by the use of softer carbons, for instance, given by r = 30 ohms, as shown by the dotted curve, II', in Fig. 178, the voltage peaks are greatly cut down, giving a lesser wave-shape cUstortion, and so, effective value of voltage, 43.1 volts, volt-amperes in arc, 395 volt-amp., watts in arc, 335 watts, hence, power-factor, 85 per cent. Comparing Fig. 178 with 177 shows that 178 fairly well approxi- mates 177, except that in Fig. 177 the second peak is lower than the first. This is due to the lower resistance, r, of the residual vapor immediately after the passage of the arc than before the starting of the arc. Fig. 177 also shows a decrease of resistance, r, immediately before starting, or after extinction of the arc, which may be represented by some expression like where 6 < 1, but which has not been considered in Fig. 178. DISTOR TION OF WA VE-SHAPE AND ITS CA USES 357 The softer the carbons, the more is the latter effect appreciable and the peaks rounded off, thus causing the curve to approach the appearance of Fig. 176, while with metal arcs, where r is very high, the peaks, especially the first, become very sharp and high, frequently reaching values of several thousand volts. Further discussion on the effect of the arc see "Theory and Calculation of Electric Circuits." 244. One of the most important sources of wave-shape dis- tortion is the presence of iron in a magnetic circuit. The mag- netic induction in iron, and therewith the magnetic flux, is not proportional to the magnetizing force or the exciting current, but the magnetic induction and the magnetizing force are related to each other by the hysteresis cycle of the iron, as discussed in Chapter XII. In an iron-clad magnetic circuit, the magnetic / ■^ \ ^ L^ ^ ^ s / ^ K \ \ \, E/ / V ^ / s s. \ \, \ \ / / / 'B \ \ \, \ / / r / \, \ \ / s. / / / \ N k \, / \ \ / f 1 y \^ \ ^ / s s \ / / V s. '^ N \ /1 1 N / \ Sk s ^ _, / ^ U / \ ^ [/ ^ ^s 1 1 Fig. 179. flux and the current, therefore, cannot both be sine waves; if the magnetic flux and therefore the generated e.m.f. are sine waves, the current \liffers from sine wave-shape, while if a sine wave of current is sent through the circuit, the magnetic flux and the generated e.m.f. cannot be sine waves. A. Sine Wave of Voltage Let a sine wave of e.m.f. be impressed upon an iron-clad reactance coil, or a primary coil of a transformer with open secondary circuit. Neglecting the ohmic resistance of the circuit, that is, assuming the generated e.m.f. as equal or practically equal to the impressed e.m.f., the voltage consumed by the generated e.m.f. and therewith the magnetic flux are sine waves, as represented by E and B in Fig. 179. The cur- 358 AL TERN A TING-C URREN T PHENOMENA rent which produces this magnetic jflux, B, and so the voltage, E, then is derived point by point from B, by the hysteresis cycle of the iron. With the hysteresis cycle given in Fig. 180, the current then has the wave-shape given as / in Fig. 179, that is, greatly differs from a sine wave. This distortion of the current wave is mainly due to the bend of the magnetic characteristic, that is, the magnetic saturation, and not to the energy loss or the area of the curve. This is seen by resolving the current wave, /, into two components: an energy component, i' , in phase with — —10 0 — r~ ^ _ — A — 8 4 0 — BV ^ / ^ j V — 4 1 0/ A — 1 -1( 10 -8 0 -6 3 U 0-2 ) ' 2 } 4 1 /e 0 80 100 + 1 J I i 0 — 0 — J '2 r Pf / — 6 y 0 — / 10 0 -B Fig. 180. the e.m.f., e = J? sin , and a wattless component, i" , in quadra- ture with E, and in phase with B. These components are calcu- lated as and where i^ and ^^_0 are the instantaneous values of the current, 7, at the angles and tt — + . . . + a„ cos n — ^5) + . . . + c„ cos (n V ./ /i 1 \ ^ r~ r/ \, s / /" / /•^ \ \ ^ \ / / / V V k. N "Nj s / / y K S s / - s N»^ / / / UJ ffl \ s \ s/ -' N / 5 3 5 n ■^ — \ \ / ^ S \ s \) / < s < 5 \| ^ . where Fig. 181. c„ = \/a„2 + 6n^, 6„ tan ^„ = a„ The coefficients a„ and 6„ are determined by the definite integrals:^ 2 r-^ a„ = - I \ TTJO IT 'Jq i cos n0d<^ = 2 X ayg (i cos n(f))o', i sin n0c?^ = 2 X a^'S' (* sin n0)o'^; that is, by multiplying the instantaneous values of i, as given numerically, by cos n0 and sin n, respectively, and then averaging. ^See "Engineering Mathematics." 360 ALTERNATING-CURRENT PHENOMENA Just as in most investigations dealing with alternating currents, not the fundamental sine wave, but the fundamental sine wave together with all its higher harmonics, that is, the total wave,, is of importance; so also when dealing with the higher harmonics, frequently not the individual higher harmonic sine wave is of importance, but the higher harmonic together with all of its higher harmonics. For instance, when dealing with the disturb- ances caused by the third harmonic in a three-phase system, the third harmonic together with all its higher harmonics or over- tones, as the ninth, fifteenth, twenty-first, etc., comes in consid- eration, that is, all the components which repeat after one-third cycle. The higher harmonic then appears as a distorted wave, including its higher harmonics. To determine, from the instantaneous values of a distorted wave, the instantaneous values of its nth harmonic distorted wave, that is, the nth harmonic together with its overtones, of order 3 n, 5 n, 7 n, etc., the average is taken of n instantaneous values of the total wave (or any component thereof, which includes the nth harmonic), differing from each other in phase by - period. That is, it is n-l o n This method is based on the relations: 2 /ex 2 « cos [vi4) -\ I = n cos m«/), 2 K^ sm vKh -\ = n sm md), \ n J if m = n or if m is a multiple of n; otherwise these sums = 0, where m and n are integer numbers. 245. In this manner the wave of exciting current, /, of Fig. 179 is resolved, in Fig. 182, into the fundamental sine wave, ii, and the higher harmonics, ^3, ^'5, i-j, which are general waves, that is, include their higher harmonics. Analytically, it can be represented by i = 8.857 cos ( is the wave of magnetic induction. DISTORTION OF WA VE-SHAPE AND ITS CA USES 361 The equivalent sine wave of above current wave is to = 9.104 cos (0 - 36.3°). In this case of the distortion of a current wave by an iron-clad reactance coil or transformer, with a sine wave of impressed e.m.f., it is, from the above equation of the current wave, Effective value of the total current 6 . 423 Effective value of its fundamental sine wave . . . 6.27 Effective value of the sum of all its higher harmonics 1 . 43. That is, the effective value of all the harmonics is 22.3 per cent, of the effective value of the total current. 11^ ^ v^ I is I \^,^ \v Fig. 182. B. Sine Wave of Current 246. If a sine wave of current exists through an iron-clad magnetic circuit, as, for instance, an iron-clad reactance coil or transformer connected in series to a circuit traversed by a sine wave, the potential difference at the terminals of the reactance cannot be a sine wave, but contains higher harmonics. From the sine wave of current i = I cos , as the current, but passes the zero much later than the current. From the wave of magnetism follows the wave of generated e.m.f., and so (approximately, that is, neglecting resistance) of JD terminal voltage, e, at the reactance, since e is proportional to --r—' It is plotted as E in Fig. 183, and resolved into its harmonics in the same manner as the current wave in A. / \ /e\ \ / , \ \ 1 ei \ ^ r" "" -- V-. ^ X / \ \ S ^-^ \ / V / >, 63 ^ y \ ^ / \ y / y / ^N \ \ (^ ^ ^ N^ / / f / "«7 k, \, \ ^/ \r ?^ =-^ .-> y- ^> / / '^ fe^ \^ \ y S2 ^ ^ ^'V ^/ \s , . X \ "^ ^ X ^ v_ 2'§ ^^ / ^ s s /- y < / / / / \ V / \ s / /' / V ^ v y Fig. 183. As seen, with a sine wave of current traversing an iron-clad reactance, the e.m.f. wave is very greatly distorted, and the maximum value of the distorted e.m.f. wave is more than twice the maximum of its fundamental sine wave. Denoting the current wave by, i = 10 sin ((/> + 30°), the e.m.f. wave in Fig. 183 is represented by e = 11.67 cos (<^ + 2.5°) + 6.64 cos 3 ((/> - 1.13'') + 3.24 cos 5 ( - 1.53°) + 1.16 cos 9 (0 - 0.5°) + 0.80 cos 11 (0 - 2°) + 0.53 cos 13 (0 - 2°) + 0.19 cos 15 (0 - 1°) + • • • that is, all the harmonics are nearly in phase with each other, so accounting for the very steep peak. It is DIS TOR TION OF WA VE-SHAPE AND ITS CA USES 363 Effective value of total wave 9.91 .Effective value of its fundamental sine wave ... 8 . 25 Effective value of the sum of all its higher harmonics 5 . 48 that is, the effective value of all the higher harmonics is 55.3 per cent, of the effective value of the total wave. The impedance of this iron-clad reactance, with a sine wave current of 7.07 effective, so is 9 91 ' = 7:07 = l-^O' while the same reactance, with a sine wave e.m.f. of 7.07 effective, in A, gives the impedance. The conclusion is that an iron-clad magnetic circuit is not suitable for a reactor, since even below saturation (as above assumed) it produces very great wave-shape distortion. As discussed before, the insertion of even a small air-gap into the magnetic circuit makes the current wave nearly coincide in phase and in shape with the wave of magnetism. C. Three-phase Circuits 247. The wave-shape distortion in an iron-clad magnetic circuit has an important bearing on transformer connections in three-phase circuits. The e.m.fs. and the currents in a three-phase system are dis- placed from each other in phase by one-third of a period or 120°. Their third harmonics, therefore, differ by 3 X 120°, or a com- plete period, that is, are in phase with each other. That is, what- ever third harmonics of e.m.f. and of current may exist in a three-p)iase system must be in phase with each other in all three phases, or, in other words, for the third harmonics the three-phase system is single-phase. The sum of the three e.m.fs. between the lines of a three-phase system (A voltages) is zero. Since their third harmonic would be in phase with each other, and so add up, it follows: The voltages between the lines of a three-phase system, or A voltages, cannot contain any third harmonic or its overtones (ninth, fifteenth, twenty-first, etc., harmonics). Since in a three-wire, three-phase system the sum of the three 364 ALTERNATING-CURRENT PHENOMENA currents in the line is zero, but their third harmonics would be in phase with each other, and their sum, therefore, not zero, it follows: The currents in the lines of a three-wire, three-phase system, or Y currents, cannot contain any third harmonic. Third harmonics, however, can exist in the Y voltage or voltage between line and neutral of the system, and since the third har- monics are in phase with each other, in this case, a potential difference of triple frequency exists between the neutral of the system and all three phases as the other terminal, that is, the whole system pulsates against the neutral at triple frequency. Third harmonics can also exist in the currents between the lines, or A currents. Since the two currents from one line to the other two lines are displaced 60° from each other, their third harmonics are in opposition and, therefore, neutralize. That is, the third harmonics in the A currents of a three-phase system do not exist in the Y currents in the lines, but exist only in a local closed circuit. Third harmonics can exist in the line currents in a four-wire, three-phase system, as a system with grounded neutral. In this case the third harmonics of currents in the lines return jointly over the fourth or neutral wire, and even with balanced load on the three phases, the neutral wire carries a current which is of triple frequency. 248. With a sine wave of impressed e.m.f. the current in an iron-clad circuit, as the exciting current of a transformer, must contain a strong third harmonic, otherwise the e.m.f. cannot be a sine wave. Since in the lines of a three-phase system the third harmonics of current cannot exist, interesting wave-shape distortions thus result in transformers, when connected to a three- phase system in such a manner that the third harmonic of the exciting current would have to enter the line as Y current, and so is suppressed. For instance, connecting three iron-clad reactors, as the primary coils of three transformers — with their secondaries open-circuited — in star or Y connection into a three-phase system, with a sine wave of e.m.f., e, impressed upon the lines. Normally, the voltage of each transformer should be a sine wave also, and equal — j^- This, however, would require that the V3 current taken by the transformer as exciting current contains a DISTORTION OF WA VE-SHAPE AND ITS CA USES 365 third harmonic. As such a third harmonic cannot exist in a three-phase circuit, the wave of maj^netism cannot be a sine wave, but must contain a third harmonic, about opposite to that which was suppressed in the exciting current. The e.m.f. generated by this magnetism, and therewith the potential difference at the transformer or Y voltage, therefore, must also contain a third harmonic, and its overtones, three times as great as that of the magnetism, due to the triple frequency. With three transformers connected in Y into a three-phase system with open secondary circuit, we have, then, with a sine wave of e.m.f. impressed between the three-phase lines, the conditions: The voltage at the transformers, or Y voltage, cannot be a sine wave, but must contain a third harmonic and its overtones, but can contain no other harmonics, since the other harmonics, as the fifth, seventh, etc., would not eliminate by combining two Y voltages to the A voltage or line voltage, and the latter was assumed as sine wave. The exciting current in the transformers cannot contain any third harmonic or its overtones, but can contain all other harmonics. The magnetic flux is not a sine wave, but contains a third harmonic and its overtones, corresponding to those of the Y voltage, but contains no other harmonics, and is related to the exciting current by the hysteresis cycle. Herefrom then the wave-shapes of currents, magnetism and voltage can be constructed. Obviously, since the relation between current and magnetism is merely empirical, given by the hysteresis cycle, this cannot be done analytically, but only by the calculation or construction of the instantaneous values of the curves. 249. For the hysteresis cycle in Fig. 180, and for a system of transformers connected in Y, with open secondary circuit, into a three-phase system with a sine wave of e.m.f. between the lines, the curves of exciting current, magnetic flux and voltage per transformer, or between lines and neutral, are constructed in Fig. 184. i is the exciting current of the transformer, and contains all the harmonics, except the third and its multiples. It is given by the equation: i = 8.28 sin ( + 30.8°) - 0.71 sin (5 0 - 17.2°) + . . . 366 AL TERN A TING-C URRENT PHENOMENA B is the magnetic flux density in the transformer. It contains only the third harmonic and its multiples, but no other harmonics, and is given by the equation: B = 10.0 sin